This sort of the variable gain amplifier circuits were hitherto implemented mostly in accordance with an approximation termed bilinear transform, represented by the following approximation (1):
                                          ⅇ                          2              ⁢                                                          ⁢              x                                ≈                                    1              +              x                                      1              -              x                                      =                  1          +                      2            ⁢                                                  ⁢            x                    +                      2            ⁢                                                  ⁢                          x              2                                +                      2            ⁢                                                  ⁢                          x              3                                +                      …            ⁢                                                  ⁢                          (                                                -                  1                                <                x                <                1                            )                                                          (        1        )            
In the bilinear transform, represented by the approximation (1), it is not ex but e2x that is approximated.
On the other hand, an exponential function is given by
                              ⅇ          x                =                  1          +          x          +                                    x              2                        2                    +                                    x              3                        6                    +          …          +                                    x              n                                      n              !                                +          …                                    (        2        )            and is represented by an identity making use of a hyperbolic function (tan h(x)):
                              ⅇ          x                =                              1            +                          tanh              ⁡                              (                                  x                  2                                )                                                          1            -                          tanh              ⁡                              (                                  x                  2                                )                                                                        (        3        )            
This identity appears in a set of formulas and may be derived with ease from the following definition of the hyperbolic function (tan h(x)):
                              tanh          ⁡                      (            x            )                          =                                            ⅇ              x                        -                          ⅇ                              -                x                                                                        ⅇ              x                        +                          ⅇ                              -                x                                                                        (        4        )            
That is, e2x may be found as
                              ⅇ                      2            ⁢            x                          =                                            1              +                              tanh                ⁡                                  (                  x                  )                                                                    1              -                              tanh                ⁡                                  (                  x                  )                                                              .                                    (        5        )            
There is not disclosed the equation (3) or (5) in past papers or Patent Documents that treat the variable gain circuits having the gain changed exponentially, with the exception of Patent Document 1 (JP Patent Kokai Publication No. JP-P2003-179447A) by the same inventor as the present inventor.
Comparing the equation (2) with the equation (1), we might readily imagine that an approximation error amounts to a rather large value.
For example, approximation of
                              tanh          ⁡                      (            x            )                          ≈                  x          +                                                    x                3                            3                        ⁢                                                  ⁢                          (                                                                  x                                                  ⁢                                                                  ⁢                                  <<                                ⁢                                                                  ⁢                1                            )                                                          (        6        )            in the equation (5) leads to the following expression for e2x:
                              ⅇ                      2            ⁢            x                          =                                            1              +                              tanh                ⁡                                  (                  x                  )                                                                    1              -                              tanh                ⁡                                  (                  x                  )                                                              ≈                                    1              +              x              +                                                x                  3                                3                                                    1              -              x              -                                                x                  3                                3                                                                        (        7        )            
The relationship between the original function e2x and its approximations (1) and (7) is shown in FIG. 1, in which the vertical axis and the horizontal axis stand for the gain (dB) and x, respectively. In FIG. 1, a characteristic (b) is for exp(2x), which is linear because it is shown in log scale. On the other hand, a curve or characteristic (a), connecting triangular-shaped dots, is for the approximation (1), and a curve or characteristic (c), connecting lozenge-shaped dots, is for the equation (7).
It will be seen from FIG. 1 that the approximation error of the bilinear transform, represented by the approximation (1), is of a noticeably large value.
It is noted that the bilinear transform, represented by the approximation (1), is to be used as an approximation for the exponential function e2x within a range of x on the order of −0.5<x<0.5.
Thus, if a circuit is to be implemented based on the approximation of the equation (1), termed the bilinear transform, the approximation error is increased. This approximation error cannot be decreased except by reducing the range of the input voltage.
[Patent Document 1] JP Patent Kokai Publication No. JP-P2003-179447A
[Non-Patent Document 1] Q.-H. Duong Q. Le, C.-W. Kim, and S.-G. Lee, “A 95-dB Linear Low-Power Variable Gain Amplifier”, IEEE Trans. Circuit & Systems-I, Vol. 53, No. 8, pp. 1648-1657, August 2006